Attempting to export as a .OBJ with .mtl, but the resultant .mtl file lacks an actual map
The problem is that the .obj exporter requires a material setup using the Principled Shader in order to write an .mtl with the map_ attributes. Unfortunately the documentation is still work in progress for Blender 2.8, that's why this wasn't previously documented.

The fix for your problems should be to replace the current shader nodes with the Principled Shader and to plug your texture in the base color socket.

Alternatively you can also add the texture by hand. This requires the following line in the .mtl if your texture is in the same directory and called pc2.dds as shown in your screenshot:

map_Kd pc2.dds